You’re unconsciously relying on Aristotle's misleading claim that: Plato is a Pythagorean. Pythagorean forms might be called schema, they answer the question about primal material. And so rely on mathematical delimitations, and with this they show their kinship to tones. One can not take Aristotle’s judgment straightforwardly, and let it impose itself, without further reflection, on our minds. What Aristotle intends to highlight is the fact that Plato, like Pythagoras, thinks that the universe is fundamentally intelligible. Plato is a Socratean. What is the concern of his teacher? It is the cardinal virtues, and the doctrine of the good, as laid down, e.g., in the Politeia. And not the question about primal material. Plato’s eidos is not the Pythagorean schema. Let us make this clear to ourselves, otherwise we will only strive in vain.

Latin schēma, from Ancient Greek σχῆμα ‎(skhêma, “form, shape”). I would propose this term for matters concerning the elemental structure of the cosmos (even though, of course, Timaeus does not use this word), i.e., that fire and water and fog and earth each were deemed insufficient to be the primal material, the primal material was thought as some vague thing, equally akin to each of the elements, until at length it had no distinct character of its own. Except this, it was not the void. But how was it to be distinguished from the void? By the theory of schema. From that followed the tonal theory.

It’s possible, but I hope not. I actually came to Aristotle through Plato and immediately realised how different his approach was.

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<<Plato is a Socratean. What is the concern of his teacher? It is the cardinal virtues, and the doctrine of the good, as laid down, e.g., in the Politeia.>>

Well, O Third Man, this is what the books tell us. But Socrates either had a wider span of interest than this, or Plato grafts his own personal interest in number on to the Dialogues. (Note how a study of number follows hard on discovery of tagathon in Republic.) If the former is true, it is Socrates who has a wider interest than the books tell of; if the latter, then Plato certainly did have this interest in harmony and number, which, by your definition, would be quite un-Socratic.

But none of this actually pertains, because if you are referring to this:

Quote:

<<Alas, if you are going to limit number to mere ‘tallying’ we’ll get nowhere. For example, why does Socrates so often give the odd and the even as twofold principles of number? Surely because they effectively ‘tame’ your infinity. Number is no longer the endless tally but the even and the odd, and thereby eidos.>>

Now the most comprehensive eide, those which come closest to the rank of an arche and are therefore termed the very first, are the odd and the even. … This “first cut” in the realm of numbers in terms of “even” and “odd” affects all numbers in such a way that “one half of the realm of number” falls under 'the odd' and the other under 'the even' each of these comprising an unlimited multitude of numbers. But each of these is now in turn gathered “into one”. GREEK MATHEMATICAL THOUGHT p.57

Klein considers all valid ways of classifying numbers to be forms, and it was on this topic that I was hoping to discourse with you a little. For example it has relevance to some of the material in the opening sections of PHILEBUS, which I believe can be rendered much clearer approaching via this route.

I am also awaiting your own thoughts on the discussion of motion. I hope your muteness on this doesn’t indicate a total cessation of thought. Motion in all its forms is, indeed, worthy of philosophic discussion. More so, perhaps , than the endless quest to find other names for the eide.

The philosopher must reject the ‘formal’, i.e., schematic use of number. Consider the musician Sergiu Celibidache who considers time the condition which brings the multitude of musical information into unity. But the information is a false condition of the narrow demand of intelligibility. In the Aristotelianism of High Scholasticism music was, so I am told, number in time. But it means number is removed from life, taken away in its counting, not one thing, not two a chair and three a table, but pure number.

If Celibidache refuses recorded music is there not something of the true Plato in that? In the Philebus this enumeration is purposed as a solution to a prior problem, which is closer to man as man. That of the one or hen and the many.

Numbers, indeed, MUST be forms, perhaps for the reason that they are intelligible. Everything must have its form. That is axiomatic. However, insofar as numbers touch on the problem of the one and the many, if this is what the Philebus is to bring before our searching eyes, I find they are not numbers at all. The one or hen is not an odd number. The many are not simply parts. Several instances of justice, i.e., just actions, can later be tallied or counted, but the odd or even of the sum would be a derivative matter concerning something somewhat alien to the matter proper. It would concern some kind of use, or pragmatic application of our intelligence about the form of Justice.

Now, it is cheap to say, Plato is Socratean. It is cheap to assign a lineage to Plato’s dearest thoughts. As he must be allowed to stand on his own feet, and one must listen to his peculiar voice. But that means the same as we who read him, must have thinking eyes, and we must find in the look of our own situation this look that goes under the rubric Plato. It can only be in becoming the things shaped therein by the text under that rubric that philosophizing would depart from schematic research and achieve its proper ken. Otherwise the philosophizing is like the recording and that formal analytical activity which is proper to the scholar. Is there then, a ground between that of the mathematicians and those absorbed by life itself? A ground of unusual elucidation, one should endeavor to induce the precious inspiration of the proper historical inquiry. We will remain genuinely bewildered, in a low and detestable way, until we come to this ground.

If Klein places the form of the division first, does he do so in speaking of his present theme, that of mathamatics? Is the arche as such indisputably best. And what is the arche? Plato is the one who broke with the traditional things. So what is original is not then to be taken in the spirit of the Chinese or the Indian thinking. For it does not point to the ancestral at all. For Plato points to nous, and the discoveries which are truly new and have all their sharpness only with the Athenian age. This arche must be properly understood.

Now, which passage, if any, from the Philebus calls for questioning? And also, are we to say more about Klein’s placement of the odd and the even. You have yet to make clear to us why you hold the subject of the odd and the even so dear.

The fact is that one is not considered a number at all within this domain of thinking, even though we might hold a finger up and expect an answer to “how many am I holding up?”. Here we standing at the threshold that transmutes arithmetike into logistike and vice versa. Socrates says the youngsters of today like to go straight from the One to infinity – neither of which are formally numbers. He wants the few to be interposed between, and in this arithmos is specified. 'The interposed' are the eide which gather the arithmoi (countable heaps) into a unity - of which the most fundamental are the even and the odd. Is this not clear? He gives the analogies of phonetics and music to help as extend to all studies. I can even extend it to guitar playing.

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<<Now, it is cheap to say, Plato is Socratean. It is cheap to assign a lineage to Plato’s dearest thoughts. As he must be allowed to stand on his own feet, and one must listen to his peculiar voice.>>

This is why I was surprised when you made your former pronouncement.

Sergiu Celibidache was an impressive conductor and there was something Socratic about his total emphasis on live performance. His philosophy is not Pythagorean, does not, as far as I know, contain any other Greek elements, and is wonderfully high blown and conductorly in the little excerpts I have heard (and which it seems he didn’t mind being recorded). If I want that sort of thing I go to Bernstein’s Harvard lectures “Whither Music…”. But all this to me seems off-ramp. We are in danger of drifting into an intellectual conversation instead of an intelligent one.

The even and the odd do need discussion in the way Klein assumes Socrates discoursed on them. Give me a day or so to see if I can make a good fist of explaining them, then you can let me know if it makes adequate sense to you. Or you could just purchase Klein's "The Origin of Greek Mathematics' (Dover Books) and find out for yourself.

Hen or the one is not odd because it is not a number at all. If I point to a chair, and say one, I use number in order to manipulate the chair. Now I can count and know how many I need. But the chair itself has nothing to do with that. It is my own counting. The forms are not thought by Plato as parts in the sense of universals that have particular countable instances. Rather, I see something that reminds me of a true chair, it is close enough as it were, but I don’t see with the sensorium the chair as one chair properly. Again, for example, when one has partial knowledge of a matter, knows something of it, but could learn more. That is not mathematical division, as if to say, he knows one third of what he is to learn about Justice.

--

Of course, if Klein will tell us something we better listen. But we must not force number, which is the theme of his little book, into Plato, who himself has said much that calls for independent and invigilating work. One is also, says Aristotle, permitted to find Cratylus at the front of Plato's paideia. Paideia, it says, one knows how to move forward in what matters.

-----

You do not see how it fits in with the most important difficulties of the Timaeus's myth. It is very instructive. One must understand it schematically first, according to the medieval examination of number. Ergo, we look with a circumlocutory pressing on.

Mathamatics, or number as a form, is neither of time nor of space. For then it is a god. When one learns first number one goes on to geometry. This is number in space, plane geometry. It is space without time. There is space for the sensorium, but no time. Music is then time without space, it is time for the sensorium, without space. The way time comes together in music to form a whole is entirely temporal, that is Celibidache’s point. It is a certain thing, not like the true music, but recalling it to the knowing animal. It has this rude time, but it is not spoiled by narrowing spatialization. (Of course, music is not sonic waves, but music.)

When you speak of math, your favoured subject, you go to the establishments of the intellect. But we must persuade you to pass away from this error. You have missed a whole range of insights. There is a terrific lesson in this understanding of music, and it is not for cultural reasons that I raise them, but for their instructive value to the thinker. Now see music without space, as the time of the sensorium (for the sensorium is not only party to measuring sense, but to the immediate divinations of what is commonly sensed). It is by divination, or immediate intuition of the essences, that one finds music and not mere noise. One collects the moments and hears a work of music. One could imagine, of course, someone who once they knew how to recognize music, all the time said, oh, when they heard, e.g., the Eight Songs for a Mad King by Peter Maxwell Davies, I know that, it is music, and so were at once done with the matter and went about their merry way. Then another who said, I only understand some hint of this music, I would learn something more of it in my lifetime, I wish a more holistic knowledge of this matter before I die. Or, one might say, this so-called music is awful. They all have divined the form, at least an aspect of it. But, their knowledge is but fragmentary, partial. The form itself has all that is to play out, for the sensorium and the intellect, it is the god of the matter in question, music.

When Goethe says that 'architecture is frozen music', he makes us a picture. Theoretically explicated his picture means that architecture is living geometry; geometry in time. On the paper where figures are drawn there is no thing at rest, things on the page are thought as being of the intellect (putting aside the matter that the paper does decay, thinking them only in their topologicality). But buildings rest, they are at the limit state of movement. Now, on the paper we understand the figures of geometry only with the intellect, the relation of the angles and of all the parts of the shapes and lines displays itself spatially, but we must think over their features and unravel their mysteries. This unraveling is from the form. But, in architecture the time belongs to the sensorium even of the very same shapes. The spatialized topos of a building stands still, it is at rest. The figures in the pure intellect do not stand still. They are not of time, but eternity.

This rest is propulsive. Not because the matter is decaying, or deporting itself towards a collapse of its structure, but for the reason that the agency in the space of buildings comes from the form, it is the proclivity of the divination of the proper use. When we say the University centers around learning, we speak of a fact. To make it more clear, the Republic is in fact of History a book about Justice, Justice is at the center of that book. But literally, one finds no powder there, at the midpoint of the book, when one opens it to about the central page, no powder called justice. So, this is a complex analogy, but it speaks of all the forms. Of their factical power. One must see that the form is not the intellect, the place of math as math, nor is it the matter of the sensorium. Its course is not directly evident. It is the impetus of the building, of the book, of the noise people call music. The building does not have an intention, it is not self aware, but it has an unconscious thinking, which is not a private possession of any person or being. No more then the sensorium’s objects are, one must have a field of sense to see, and one must be ‘human’ to receive these divinations.

One may also mention that, mathematicians have never been proclaimed the guardians of Justice, and it would be unusual if anyone were to purpose it. The verbal definitions Socrates purposes are of the same ken as the mathematical proofs, both have forms, but neither are the basis of form as such. They are chiefly locators or indicators, which offer human beings guidance.

<<Mathamatics, or number as a form, is neither of time nor of space. For then it is a god. When one learns first number one goes on to geometry. This is number in space, plane geometry. It is space without time. There is space for the sensorium, but no time. Music is then time without space, it is time for the sensorium, without space. The way time comes together in music to form a whole is entirely temporal, that is Celibidache’s point. It is a certain thing, not like the true music, but recalling it to the knowing animal. It has this rude time, but it is not spoiled by narrowing spatialization. (Of course, music is not sonic waves, but music.)>>

Let me ask you, O Third Man, a single question, which I consider germane to our discussion and a safe entry into the-subject-that-without-care-is-most-of-all-likely-to-cascade-out-of-control, music.

<< Hen or the one is not odd because it is not a number at all. If I point to a chair, and say one, I use number in order to manipulate the chair. Now I can count and know how many I need. But the chair itself has nothing to do with that. It is my own counting. The forms are not thought by Plato as parts in the sense of universals that have particular countable instances. Rather, I see something that reminds me of a true chair, it is close enough as it were, but I don’t see with the sensorium the chair as one chair properly. Again, for example, when one has partial knowledge of a matter, knows something of it, but could learn more. That is not mathematical division, as if to say, he knows one third of what he is to learn about Justice.>>

Well, agreed. When you say ‘three chairs’ you don’t seem to be counting forms, but appearances. “Pure” forms aren’t counted, are they? Aristotle says that the form is like a seal, masculine and singular in that it is pattern - pater, whereas matter or mater, being feminine, is that which is patterned or stamped with the seal, and which is manifold (or it could be only one). Either way the first is eidos and the second, mathematically speaking, concerns units. Units are not themselves arithmoi until they are counted, only then do they enter the kingdom of the odd and even. So it may be more intriguing than at first strikes us. Uncounted units are the Indefinite (regulated the ancient mathematicians would say, by the dyad). If units are counted, they become arithmoi. The last number of the tally is the arithmos is what Klein says. But that’s his reading of the affair. It seems to me that there is nothing to stop you ‘counting’ a single chair and calling it ‘one’, but it will be a chair of the sensorium not of nous; and you are performing a dianoic operation, not a noetic. You do not count the form.

There are more interesting properties concerning the even and the odd, but not all at once, eh. We have to prove our maturity of vision. I don’t see this as interesting unless one's interest in number is originative and one is sensitive to subtle distinctions - that is, understand that these matter terriby. Such distinctions must be honed and cultivated if they are to transfer to the study of music. Tallying has no place in harmony, except accidentally.

Therefore this turns out not to be the subject one first thinks it is.

<<One could imagine, of course, someone who once they knew how to recognize music, all the time said, oh, when they heard, e.g., the Eight Songs for a Mad King by Peter Maxwell Davies, I know that, it is music, and so were at once done with the matter and went about their merry way. Then another who said, I only understand some hint of this music, I would learn something more of it in my lifetime, I wish a more holistic knowledge of this matter before I die. Or, one might say, this so-called music is awful.>>

Although what you say here purports to be about music, you really talk of judgment, defunct or evolving, and its effect on the mind; you could be talking of not only any artefact but also any spectacle of nature. The same phrase, “I know that, it is [something]” could be said of anything that comes under perception and judged. Therefore you do not bring us near to music only towards an aspect of sequential disclosure – e.g. a dramatic performance or even a rugby match. From such a point it is tricky to begin to move to ‘the thing itself’. One such beginning would be to respond to my original question: Is time associated with music linear or cyclic?

Another last point of reflection is Plato/Socrates remark in Republic:

“The just man will not allow the three elements which make up his inward self to trespass on each other’s functions or interfere with each other, but by keeping all three in tune, like the notes of a scale (high, middle, and low, and any others there be), will in the truest sense set his house to rights, attain self-mastery and order, and live on good terms with himself.” [443d]

This analogy of Socrates, I believe, bears some relation the musical order of the Lamda numbers in Timaeus. From these numbers and their expansions, translated to string-lengths, we obtain what Plato called the true Hellenic mode. And these numbers, based on powers of 2 and 3, gather within the limit of 27, a tellingly divined number in that it is the sum of its constituent parts (1+2+3+4+8+9=27 - a fact that must have brought a mischievous smile to Plato’s face).

Are you sure? Because it seems that the one who wants to learn more is driven on towards the whole. And not the others, who are already satisfied or who sneer. Of course it is true, here one says nothing that couldn’t be said about any matter.

This that you call 'judgment' need not be counted a judgment, judgment implies active reasoning, that a proposition, 'that is music', taken as an immediate intuition of the character of what stands there, be interrogated in, e.g., the style of Socrates. Or, again, in the western-modern style of the logics of inference and validity (according to the doctrine of event-effects & event-causes, and not of natural character).

It is possible to understand these matters, perhaps, if we radically extirpate the modern sense of the word math (Descartes, Frege, et al), and thereby, consider it as an art, self sufficient and involving nothing beyond itself.

Goethe says, taken losley in the translation: the Mathematician is most perfect when he has the paideia which concerns the beauty of the true. The paideia, with respect to math, is the sense of what matters most in math. Therefore the self-guidance, or inner genius of the mathematician is guided towards the beauty of the true. But of course, this math is not the math of Descartes and Frege. It is the object of the paideia as what is apprehended by the mathematician and inscribed in math. Math itself has no object, in the sense that chemistry studies compounds, mathematics invents numbers out of whole cloth, and various maths have different senses of number. Likewise the true music can not deal with tones, but with the relations that belong to music as music, and without an external object. The music itself is the object, it is the meaning made available to the others, where meaning arises as a transformation of time into music.

What I would say first is that music is not like the sciences that have objects, external to themselves, at there center. But like math it is an art, distinguished form, e.g., physics, which has objects external to it own nexus and formations.

The time involved in the composition is sometimes called linear for the reason that it is not addressed to eternity. But in the fulfilled fullness of a moment which is only itself, and has no character but that of the inner life (but not the subjective biography) of the composer who is addressing what matters most, must absorb himself in the possibilities of musical composition, and in producing his composition he has brought his time into the field of music which is available to others, the linear time of Zeno (which is not cyclical like the ‘time’ of the will of the eidoi). The higher dignity of such a work depends in some way on the development of the possibility for such a composition, and thus is an historical achievement and the deeper the powers of the composer, the greater the meaning which suffuses the historical world of which he belongs. Because the interpretation of the vague and unformed comes into aprehensible being in the linearity of the time of change and destruction.

The forms, in the music, would not be linear, as forms. But that is true of everything. It is perhaps that what such an art does, is play the forms like a remote organ. Ergo, the will of nature, to which all belong, is intuited.

Goethe says, taken losley in the translation: the Mathematician is most perfect when he has the paideia which concerns the beauty of the true. The paideia, with respect to math, is the sense of what matters most in math. Therefore the self-guidance, or inner genius of the mathematician is guided towards the beauty of the true. But of course, this math is not the math of Descartes and Frege. It is the object of the paideia as what is apprehended by the mathematician and inscribed in math. Math itself has no object, in the sense that chemistry studies compounds, mathematics invents numbers out of whole cloth, and various maths have different senses of number. Likewise the true music can not deal with tones, but with the relations that belong to music as music, and without an external object. The music itself is the object, it is the meaning made available to the others, where meaning arises as a transformation of time into music.

I think I like this, O Third Man, but I won't know until until you say a little more. For you are still talking 'of' and until you give me a single example this might be just something you have read.

Quote:

Math itself has no object, in the sense that chemistry studies compounds, mathematics invents numbers out of whole cloth, and various maths have different senses of number.

Yes, this needs to be made real. It is not strictly true that maths has no object, though it is certainly true that it has different objects than chemistry, or even physics, and certainly primitive farming.

What I am tempting you to do is show me that you can see this 'as it happens'. Spread something out before us that doesn't so much interpret what you are saying about 'the sense of what matters most in math' but lays it out simply in a way that we can see it simply. Surely that would be the Greek way.

If you can't that's fair enough, it's not an easy call, but then the onus is on me to do the deed.

“Math itself has no object, in the sense that chemistry studies compounds, mathematics invents numbers out of whole cloth, and various maths have different senses of number.”

“Yes, this needs to be made real. It is not strictly true that maths has no object, though it is certainly true that it has different objects than chemistry, or even physics, and certainly primitive farming.”

Let us not be glib. ‘It is not strictly true that maths’ have no object. I don’t see that. A stone is the object named. A crosswalk is the thing expressed or made external by the lines on the ground. Justice is what is divined, or ‘partially known’ in some rather dark way (i.e., it is Socrates who says, I don’t know what virtue is Meno, how then can I say if it is or is not teachable?). Math and language goes one step further, it tends towards what Kant called the sublime, it ‘breaks the sphere’ in its unhealthy infinity (it seems to controvert the perfection of Justice). Of course, one can point to an instance of justice, a just action, but that does not make justice like a stone. I.e., it is not subject to historical examination in the sense of true inquiry. I.e, in the sense of looking at that stone with, e.g., a microscope.

Here we have shown a step into the essence of music, as a differentia. But this differentia belongs also to number and word.

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“I think I like this, O Third Man, but I won't know until until you say a little more. For you are still talking 'of' and until you give me a single example this might be just something you have read.”

The example would give us a particular belonging to the ‘talking ‘of’’ point. That is not what is meant here. Not the experience. The ‘sleep-inducing properties of opium’ the ‘power’ behind the thing as it is, point to the truth of the essence of the opiate. I.e., to the thing divined. The thing divined has nothing to do with the stuff we would need to make that thing, or the knowledge of making it, but with the factical ground, or meaning. If we had no meaning at all, the knowledge of the general thing, and the ability to make it, would stand in oblivion. We, today, with our characteristic cheapness, pass over this issue, smugly. We have become careless, and it is no longer quite intelligible to us why the ancients to such an extent took so strict a view of laughter. Why they resisted it. Only the how-to seems real to us, while the element itself becomes laughable.

This can not be put down, but only situated by the suggestive power of language. It is not to be experienced by 'this one', or by them, but it is the raising up of the divining to the sight of the human being. To make issue of it.

addendum
In listening to Searle's position one experiences something of the current nonsense:

Searle says: idealism is the view that reality is mental, states of consciousness actual and possible,
[Searle has no training in Greek thought to speak of, speaks of mind and psuke as though they were synonyms]

The preposterous idea that the psuke, which is the sensorium, is opposed to the physical. The psuke means imagination, perception, and representation. Representation is the given thing itself, the idea. The thing there, a stone, a copy of Husserl's Logical Investigations.

People presenting themselves as teachers of philosophy can not even see that for the Greeks the physical is opposed to the true, and not to the mental? This is the actual state of philosophy today. Even the blindest amongst us can see that if such a man has been praised as a philosopher for over 50 years, and has never been corrected in his basic defects, the diminished state of all thought.

It is not as though Searle were stupid. He is, rather, intelligent and able. We should not evade the seriousness of the diminishment by pretending it is some personal defect. Such a man would have been put to good use prior to the confusion, and to the rise of the current tendency. One can say the same of others, e.g., Derrida and Foucault. As it is, there is nothing of the kind that can be called serious philsophy these days.

The situation does not improve when we go into the innumerable non-famous specialists in Aristotle and Plato, for instance, or even in Husserl or Dilthy. On the contrary, the incurability seems all the more evident, the tendency even more and more deliberately empty, the experience of auditing these people is of a void with no end, and empty babble about orginal findings, which turn out to be nothing but mechanical explorations of already evident combinatory possibilities.

One should let this finding, be taken up concretely, as what is to motivate the hard-dealing breaking of the binding snare of the crude-ideological and methodological imbecility.

The preposterous idea that the psuke, which is the sensorium, is opposed to the physical. The psuke means imagination, perception, and representation. Representation is the given thing itself, the idea. The thing there, a stone, a copy of Husserl's Logical Investigations.

What is interesting for the forum in Berkeley is that he put nous (through the character Philonous) and dianoia (through the character Hylas) on trial (as in the upper part of the analogy of the line). And argues that our knowledge of the world must come through nous, through knowing. Today they call it 'conscious' knowing. Consciousness is what accompanies every sense data as a selective apperception of the psuke, and a bringing into the secret life of the memory of the being.

What he shows is that dianoia is a theoretical addition to nous . Thus Plato teaches that nous is the direct opening of beings onto their world, whereas diaonia is the shadow of nous.

Nous itself grasps what it grasps according to the tendency of the current psuke, the life of the being now here. It thus involves a prejudice. Nous is not seen by Plato as the perfect nous, but rather it points to the positive potential to reach that perfect knowing.

What is of further and decisive interest is that if even thought that is but several hundred years old is so muddled, how much more suspicion should one have about the translation of ancient texts? It does not help at all that one reads the Greek directly, one reads a translation (only skipping the step of writing it out).

Likely one knows nothing of Plato. One would have to have the same nous and the same tendency of bringing to light what was there. To use a metaphor, the humble ant has a psuke, a life of its own, but it does not open out its gaze so far as to see Husserl's Logical Investigations. It brings out from the clearing what its psuke has already begun to drive towards. Even the ant has not come to its proper and true life, but even more and more it opens the clearing.

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